Nit-Picky Questions on Induced EMF, and Electromagnetism in General

[pr33ch]

I just need some clarification on the induced EMF Formula.. I know that the negative sign in front of N * dFlux/dt is just an indicator of Lenz's law, but if we were given the magnetic flux as a function of time, would there be situations where -d/dt(F(t)) would be different than just d/dt(F(t))?

Also -- this is a more conceptual question -- when you think about how fields and forces are related in electrostatics, the electric field and mechanical force vectors are parallel. Before learning about the right hand rule, I intuitively thought that magnetic fields would behave in the same manner with regards to the mechanical force, but obviously it doesn't. So my question is, what properties of magnetic fields make it so that the mechanical force is perpendicular to both the field lines and the direction of the moving charge?

One more point that I really need clarification on is the relationship between electricity and magnetism. I produces a B, which produces F. Also changing magnetic flux produces ε which produces I. So basically magnetism is electricity in motion, and electricity is magnetism in motion. However, I do not understand exactly why these concepts hold true - I don't want to simply parrot information -- I'd like to understand and visualize how these concepts work. I'd love a detailed explanation!

Sorry about the long post... these questions have just been bothering me for quite a while now.

Homework Statement

Concepts and application of a derivative regarding induced EMF

Homework Equations

RHR 1 + 2, Induced Emf = -N * dFlux/dt, F = qVBsin(θ) = ILBsinθ

The Attempt at a Solution

 

[tiny-tim]

welcome to pf!

hi pr33ch! welcome to pf!

(i'll leave it to someone else to answer your first paragraph)

… what properties of magnetic fields make it so that the mechanical force is perpendicular to both the field lines and the direction of the moving charge?

at least two answers:

i] anything that can happen mathematically will happen physically …

since that relation between force and motion is possible, it would be surprising if it wasn't found in nature

ii] the (pseudovector) magnetic field comes from a vector potential (as opposed to the scalar potential of an electric field), and that's how a vector potential works

I produces a B, which produces F. Also changing magnetic flux produces ε which produces I. So basically magnetism is electricity in motion, and electricity is magnetism in motion. However, I do not understand exactly why these concepts hold true - I don't want to simply parrot information -- I'd like to understand and visualize how these concepts work. I'd love a detailed explanation!

again, at least two answers, a rather technical one first:

i] The electric (vector) and magnetic (pseudovector) fields, E and B, are really part of a single electromagnetic field, the 2-form (E;B).

[itex](\boldsymbol{E};\boldsymbol{B})[/itex] is the boundary of a 1-form (a vector): [itex](\boldsymbol{E};\boldsymbol{B})\ =\ \partial\wedge(\phi,\boldsymbol{A})[/itex], and the boundary of a boundary is 0: [itex]\partial\wedge\partial\wedge\ =\ 0[/itex]

As a result, it is impossible to have a changing magnetic field without a rotational (non-conservative) electric field: this is the Maxwell-Faraday equation: ∂B/∂t + ∇ x E = 0, one half of (E;B) = 0.

When we control the electric field, we say it generates the magnetic field. When we control the magnetic field, we say it induces the electric field. But in reality they go together: they are essentially two sides of the same coin.

Any electric field [itex]\boldsymbol{E}[/itex] is the sum of an irrotational part [itex]-\boldsymbol{\nabla}\phi[/itex] (with a scalar gradient [itex]\phi[/itex]), whose curl is 0, and a rotational part [itex]-\partial\boldsymbol{A}/\partial t[/itex] (with a vector gradient [itex]\boldsymbol{A}[/itex]), whose curl is [itex]-\partial\boldsymbol{B}/\partial t[/itex]:

[itex]\boldsymbol{E}\ =\ -\boldsymbol{\nabla}\phi - \partial\boldsymbol{A}/\partial t[/itex]
[itex]\boldsymbol{\nabla}\times\boldsymbol{\nabla}\phi\ =\ 0[/itex]
[itex]\boldsymbol{\nabla}\times\boldsymbol{E}\ =\ -\boldsymbol{\nabla}\times \partial\boldsymbol{A}/\partial t\ =\ -\partial(\boldsymbol{\nabla}\times \boldsymbol{A})/\partial t\ =\ -\partial\boldsymbol{B}/\partial t[/itex]

ii] electric and magnetic fields ultimately are generated by the ordinary Coulomb 1/r² radial purely electric field of stationary charges

when we start moving those charges, we can fairly easily work out (from transformation symmetry arguments) that that purely electric field acquires a velocity-dependent effect

(eg, look at a test charge accelerating in a straight line radially from an infinite line of stationary charge … in a moving frame, that's moving charge, and the test charge is now following a curved line, thus showing that there's a perpendicular component to the force!)​

if we use Newtonian relativity, we get most of the relationship

if we use einsteinian relativity, we also get the full maxwell-faraday equation

so the connection between electricity and magnetism is actually an obvious consequence of Coulomb's law! ​

 

[pr33ch]

Thank you very much - Both for the help and the warm welcome ^_^

That very last conceptual explanation seriously helped me out a lot, and I don't know why I didn't notice that since the beginning! But I'm honestly not familiar with vector potentials, and that's probably what's hindering my ability to understand the derived equations